2010/05/26 by Kenichi Bannai, Shinichi Kobayashi · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Mathematical Identities #Advanced Algebra and Geometry
paper · doi:10.1215/00127094-2010-024
openalex publication_date 2010/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-functions of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (“normalized” or “canonical” in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic properties of reduced theta functions for abelian varieties with complex multiplication (CM). As a corollary, when the prime p is ordinary, we give a new construction of the two-variable p-adic measure interpolating special values of Hecke L-functions of imaginary quadratic fields, originally constructed by Višik-Manin and Katz. Our method via theta functions also gives insight for the case when p is supersingular. The method of this article will be used in subsequent articles to study in two variables the p-divisibility of critical values of Hecke L-functions associated to imaginary quadratic fields for inert p, as well as explicit calculation in two variables of the p-adic elliptic polylogarithms for CM elliptic curves